Cooling timescale of dust tori in dying active galactic nuclei

We estimate the dust torus cooling timescale once the active galactic nucleus (AGN) is quenched. In a clumpy torus system, once the incoming photons are suppressed, the cooling timescale of one clump from $T_{\rm dust}=1000$ K to several $10$ K is less than $10$ years, indicating that the dust torus cooling time is mainly governed by the light crossing time of the torus from the central engine. After considering the light crossing time of the torus, the AGN torus emission at $12~\mu$m becomes over two orders of magnitude fainter within $100$ years after the quenching. We also propose that those "dying" AGN could be found using the AGN indicators with different physical scale $R$ such as $12~\mu$m band luminosity tracing AGN torus ($R \sim 10$ pc) and the optical [OIII]$\lambda5007$ emission line narrow line regions ($R=10^{2-4}$ pc).